title: Deterministic and stochastic bifurcations in the Hindmarsh-Rose neuronal model.
abstract: We analyze the bifurcations occurring in the 3D Hindmarsh-Rose neuronal model with and without random signal. When under a sufficient stimulus, the neuron activity takes place; we observe various types of bifurcations that lead to chaotic transitions. Beside the equilibrium solutions and their stability, we also investigate the deterministic bifurcation. It appears that the neuronal activity consists of chaotic transitions between two periodic phases called bursting and spiking solutions. The stochastic bifurcation, defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value, or under certain condition as the collision of a stochastic attractor with a stochastic saddle, occurs when a random Gaussian signal is added. Our study reveals two kinds of stochastic bifurcation: the phenomenological bifurcation (P-bifurcations) and the dynamical bifurcation (D-bifurcations). The asymptotical method is used to analyze phenomenological bifurcation. We find that the neuronal activity of spiking and bursting chaos remains for finite values of the noise intensity.
authors.affiliation: Laboratory of Mechanics and Materials, Department of Physics, Faculty of Science, University of Yaoundé I, Box 812, Yaoundé, Cameroon.
authors.forename: S R
authors.initials: SR
authors.lastname: Dtchetgnia Djeundam
authors.forename: R
authors.initials: R
authors.lastname: Yamapi
authors.forename: T C
authors.initials: TC
authors.lastname: Kofane
authors.forename: M A
authors.initials: MA
authors.lastname: Aziz-Alaoui
mesh_terms: D000200:Action Potentials / Q000502:physiology; D000465:Algorithms; D000818:Animals; D001703:Biophysics / Q000379:methods; D003198:Computer Simulation; D006801:Humans; D008959:Models, Neurological; D009415:Nerve Net; D009474:Neurons / Q000378:metabolism / Q000502:physiology*; D017711:Nonlinear Dynamics; D016011:Normal Distribution; D011336:Probability; D013269:Stochastic Processes; D013997:Time Factors